# Kodaira classification and the McKay correspondence

Kodaira's table of singular fibers has a singular fiber for each of $\tilde{A}_n$, $\tilde{D}_n$, $\tilde{E}_6$, $\tilde{E}_7$, and $\tilde{E}_8$; these are chains or cycles of (-2)-curves connected to each other based on the aforementioned Dynkin diagram.

On the other hand $A_n$, $D_n$, $E_6$, $E_7$, and $E_8$ Dynkin diagrams, by McKay correspondence, describe the exceptional curves of quotient singularities; i.e. if $G$ is a finite group corresponding to one of these Dynkin diagrams, $\mathbb{C}^2/G$ has a resolution whose exceptional curve is equal to that Dynkin diagram.

Question: Is there a similar construction of Kodaira's singular fibers? i.e. can we obtain them as a resolution of some quotient prescribed by one of $\tilde{A}_n$, $\tilde{D}_n$, $\tilde{E}_6$, $\tilde{E}_7$, and $\tilde{E}_8$.

If you take the Tate curve $\mathbb G_m/\langle q \rangle$ over $\mathbb Z[[q]]$ and mod out by the subgroup $\mu_n$ and resolve singularities, you should get a copy of the semistable fiber type $I_n$ with lattice $A_{n-1}$. Note that this means the Tate curve, modulo nothing, is of fiber type $I_1$, and hence already has a singular fiber.
However $I_n$ can also be expressed in a number of other ways, like as a cover of $I_1$.
The fibers $I_n^*$ can be expressed as the quotient of $I_{2n}$ by the automorphism sending $q \to -q$ and inverting $\mathbb G_m$. So combining this with the previous we can write it as a quotient of $I_1$ by a $D_{2n}$ subgroup, or $I_2$ by a $D_n$.
The remaining fibers can be viewed as resolutions of a quotient of a family of smooth elliptic curves. One takes a constant elliptic curve $E$ over a curve with coordinate $t$ and an automorphism $\sigma$ of $E$ of order and takes the quotient by the map $E \to \sigma(E)$, $t \to \mu t$ for $\mu$ a fixed $n$th root of unity. The different automorphisms and roots of unity will give all the remaining fiber types.